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1 SCHOOL OF FINANCE AND ECONOMICS UTS: BUSINESS WORKING PAPER NO. 46 March 2006 Keynesian Disequilibriu Dynaics: Convergence, Roads to Instability and the Eergence of Coplex Business Fluctuations Pu Chen Carl Chiarella Peter Flaschel Hing Hung ISSN:

2 Keynesian Disequilibriu Dynaics: Convergence, Roads to Instability and the Eergence of Coplex Business Fluctuations Pu Chen Faculty of Econoics Bielefeld University Gerany Carl Chiarella School of Finance and Econoics University of Technology, Sydney Australia Peter Flaschel Faculty of Econoics Bielefeld University Gerany Hing Hung School of Finance and Econoics University of Technology, Sydney Australia March 7, 2006 Abstract: We reforulate the traditional AS-AD growth odel of the Neoclassical Synthesis (stage I) with a Taylor policy rule replacing the conventional LM-curve, with gradually adjusting wages as well as prices, and with perfect foresight on current inflation rates and an adaptively revised notion of an inflationary cliate in which the econoy is operating. We copare this approach with the New Keynesian approach, the Neoclassical Synthesis, stage II, with staggered price and wage setting and find various coon coponents, yet with radically different dynaic iplications due to our treatent of the forward-looking part of our wageprice spiral. We show for a syste estiate of our odel that it iplies qualitatively local asyptotic stability and when its estiated for is siulated in response to isolated shocks strongly daped business fluctuations, due to a stable interaction of goods arket dynaics with the interest rate policy of the central bank and due to a noral working of a real-wage feedback chain. These results are however endangered leading in fact to econoic breakdown when there is a global floor to oney wage inflation rates. In this case, the return of soe oney wage flexibility in deep depressions is of help in restoring viability of the odel, thereby even avoiding explosive dynaics and the collapse of the econoy. This situation leads to viable, but coplex business fluctuations. Keywords: DAS-DAD dynaics, wage and price Phillips curves, real interest effects, real wage effects, (in)stability, persistent business cycles, coplex dynaics. JEL CLASSIFICATION SYSTEM: E24, E3, E32.

3 . Introduction In this paper we continue the analysis of an epirically oriented baseline odel of D(isequilibriu)AS-D(isequilibriu)AD type that we initiated in Chen, Chiarella, Flaschel and Seler (2006). This odel type was designed as an alternative to the New Keynesian odel with both staggered price and wage setting. Its origins as far as the considered wageprice spiral is concerned date back to the paper of Chiarella and Flaschel (996). This wageprice echanis has recently been extended and studied analytically and nuerically in Chiarella, Flaschel and Franke (2005) as well as in the above paper by Chen et al. The results we obtain fro this echanis, augented by (partly) conventional Keynesian goods arket dynaics, Okun s law and a conventional type of Taylor interest rate policy rule stand in striking contrast despite foral siilarities to the ones obtained fro the coparable New Keynesian acrodynaics as will be shown in this paper. We use estiated paraeter sets to study the stability features of our odel nuerically as well as analytically. The strong convergence properties that we will obtain in this way -- for a onetary policy that is sufficiently active -- however only apply to the by and large linear version of our DAS-DAD syste. They will copletely disappear (in the downward direction) when this odel type is enhanced by downward wage inflexibility, since the estiated DAS-DAD odel is profit-led (where real wage increases are contractionary due to a doinance of investent over consuption in this situation). In a profit-led econoy with downwardly rigid wages and (aybe only sluggishly) falling prices, we obtain in a depressed situation the outcoe that real wages ust increase, a fact which will ake the existing depression ever deeper, until the econoy either collapses or undergoes a significant change in behavior. This real wage channel as well as ore conventional channels (known under the heading Keynes- and Mundell-effects) will be studied in detail fro the nuerical and the analytical perspective. Finally we will consider the case where the econoy is wage-led, where increasing wage flexibility is bad for econoic stability, since boos will tend to produce real wage increases which then further stiulate the econoy under such a regie. The sae accelerator echanis is of course then working in a downward direction, but can obviously (fro a partial perspective) be stopped if there are floors to oney wage deflation (or even inflation, as was indeed shown to be the case in Hoogenveen and Kuipers (2000) for six European countries). In a wage-led regie a kink in the oney wage Phillips curves therefore can liit the purely explosive behavior that exists in the unrestricted case and can indeed be shown to lead even in a fairly advanced 5 diensional syste to bounded and in fact econoically viable dynaics which oreover will be of a coplex type if the unrestricted dynaics becoes strongly explosive. We thus in su find interesting dynaical features in an advanced Keynesian aggregate supply / aggregate deand odel with sluggish price, wage and output adjustents that allow for convergence results (in particular when estiated) and thus for considerations that relate to the Frisch paradig in business cycle theory. However, not totally unrelated to the estiated sizes of paraeter values, we can also find situations where endogenously generated irregular business fluctuations are observed, which in our view confir (though with tie-invariant paraeters still), the cycle theory advanced in Keynes (936) General Theory, which we would characterize as an anti-frisch or Keynes paradig. Nothing of this sort however holds for the New Keynesian odel in its deterinistic continuous tie core version for which we will find in this paper that the coparably general odel (with both

4 staggered prices and wages) is copletely indeterinate even for active onetary policy rules, in striking contrast to its baseline version that is norally considered in the theoretically oriented literature. 2. Baseline DAS-DAD acrodynaics In this section we reconsider a odel of the D(isequilibriu)AS-D(isequilibriu)AD variety, introduced in Chen, Chiarella, Flaschel and Seler (2006) as an epirically otivated reforulation of a baseline odel of the Keynesian D(isequilibriu)AS-AD variety investigated in Asada, Chen, Chiarella and Flaschel (2006). These odel types can be characterized as atured redesigns of the standard odel of the old Neoclassical Synthesis and will be contrasted with the corresponding odel type of the now fashionable New (Keynesian) Neoclassical Synthesis. In our baseline odels with their dynaic or static forulations of goods arket behavior we avoid the logical inconsistencies of the old Neoclassical Synthesis, described in detail in Asada, Chen, Chiarella and Flaschel (2006), basically by forulating a wage-price spiral echanis consisting of a oney wage Phillips curve (WPC) and a price Phillips curve (PPC) that at first sight look very siilar (fro the foral perspective) to the wage-price dynaics of the New Keynesian odel, when both staggered prices and wages are considered in the latter approach. We have qualitatively seen the sae variables and the sae paraeter signs on the righthand sides of these two PC s (as far as the dependence of wage and price inflation on output and wage gaps are concerned), but use as in our earlier work on Keynesian acrodynaics, see Chiarella and Flaschel ((996),) for a first forulation, always hybrid inflationary expectations foration for the accelerator ters we eploy. In our forulation of wage and price inflationary expectations foration, we have on the one hand yopic perfect foresight, not as in the case of the New Keynesian wage-price dynaics on the own one-period-ahead rate of inflation (a self-reference echanis), but with respect to other rate of inflation (a hetero-reference echanis), as it is appropriate when one speaks of cost-pressure ites in the tradition of ainstrea Phillips curve forulations. Due to this cross-over structure in the yopic perfect foresight coponent of PC accelerator ters we can furtherore assue a neoclassical dating of these expectations, eaning thereby that tie indices of inflation rates are the sae on both sides of these two PC s, whereas the New Keynesian Phillips Curves use the current rate and the one-period-ahead wage inflation (or price inflation) rate on the leftand right-hand sides of their staggered wage (respectively price) adjustent rules. In addition, we have always backward looking expectations included, but interpret such (adaptively updated) expectations as an inflation cliate expression that agents for in addition to their correct yopic expectations, in order to also pay attention to the inflationary regie into which current inflation is ebedded. In coparison to the New Keynesian wage-price dynaics, see Woodford (2003) for exaple: = = dln wt Et( dln wt+ ) + βwyyt βw ω lnω t dln p E( dln p ) + β y+ β lnω t t t+ py t pω t y ln t = Yt the log of current output and d, the backward difference operator. Note also that we ignore in these equations the discount paraeter β, β of the New Keynesian approach by setting it equal to one for expositional siplicity.

5 we instead ake use of the following representation of a Keynesian wage-price spiral (still using the New Keynesian easures for the output and the wage gap, however): The structural for of the Keynesian wage-price spiral (in discrete tie): WPC t+ κw t t+ + κw t + βwy t wω = dln w E ( dln p ) ( ) lny β lnω t PPC t+ κp t t+ + κp t + βpy t+ βp ω t = dln p E( dln w ) ( ) lny lnω where denotes our inflationary cliate expression, here updated by a standard adaptive expectations process in order to siplify the analysis of the odel. Up to expectations we thus have the sae foral structure in these wage-price dynaics, while expectations are now based on weighted averages of corresponding cost pressure ters, cobining yopic perfect foresight with sluggishly adjusted inflation regie expectations. The difference between the New Keynesian and our approach oreover lies in different icrofoundations of the wage and price PC s, based on what has been shown in Blanchard and Katz (999), besides the above significantly different way of treating forward- and backward-looking expectations. Transferred into a deterinistic continuous-tie fraework, the New Keynesian wage-price w p dynaics reads (, their wage and price inflation rates): = w βwy y + βw ωθ p = β py y βp ωθ while our approach now specifically in ters of the eployent rate e on the labor arket 2 and the capacity utilization rate u on the arket for goods then gives rise to: The structural for of the Keynesian wage-price spiral (in continuous tie): w wˆ = = β ( e ) β ln ω+ κ pˆ + ( κ ), κ (0, ) we wω w w w p pˆ = = β ( u ) + β ln ω+ κ wˆ + ( κ ), κ (0, ) pu pω p p p () (2) The difference between our and the New Keynesian approach to wage price dynaics therefore is that we use on the left hand sides inflation rates wp ˆ, ˆ for wage and price inflation in place of their tie rate of change, use two easures for the output gap, one relating to the labor arket (the deviation of the rate of eployent fro the NAIRU rate of eployent e = ) and one relating to the goods arket (the deviation of the rate of capacity utilization of firs fro their intended noral rate of capacity utilization u = ), and in both the WPC and the PPC weighted averages for the cost-pressure easures of both workers and firs, based on yopic perfect foresight and our concept of an inflation cliate. Due to these difference in wage and price inflation foration, we do not get the sign reversal in front of the output and wage gaps that is typical for the New Keynesian approach to wage and price dynaics. The WPC has been icrofounded in Blanchard and Katz (999) fro the perspective of current theories of the labor arket. Note that the log that appears in the foral representation 2 with ˆx = xx / the growth rate of a variable x. Note also, that as in the New Keynesian approach we have noralized steady states rates, now of eployent rates and capacity utilization rates of firs and also for the real wage to.

6 of our WPC is not due to a loglinear approxiation of the originally given structural equation, but instead results, see Blanchard and Katz (999), fro a growth rate reforulation of an initially in level for represented bargained real-wage wage curve. For the PPC a forally siilar procedure is available, based on an elaborate approach to flexible arkup pricing. The two PC s can be considered as a linear syste of equations in the variable wˆ, pˆ that can be easily solved, giving rise thereby to the law of otion (6) for the real wage ω = w/ p and the reduced for price PC (8). Together with the law of otion for the inflationary cliate expression (7) we therefore obtain in su three laws of otion that describe the Disequilibriu adjustent of Aggregate Supply of our odel (the DAS-coponent) as is shown below. The Disequilibriu AD part of the odel (the DAD coponent) is here of a siple type still, consisting of a dynaic ultiplier equation (3) in ters of the rate of capacity utilization u, whose rate of growth is here assued to depend negatively on its level u (as in the siple textbook ultiplier story), as usual on the (here actual) real rate of interest r pˆ and, with abiguous sign, on the real wage, which easures the ipact of incoe distribution on the deterination of aggregate deand and resulting output adjustents. We call a regie where α uω applies a profit-led regie and the opposite case a wage-led regie, characterizing in this way the situations where investent doinates consuption with respect to real wage changes and vice versa. Adding to this law of otion for the rate of capacity utilization, we assue that the rate of eployent e is obeying soe sort of Okun s law, following the rate of capacity utilization with a tie delay as shown in equation (4). We finally have a standard for of a Taylor interest rate policy rule (5), including however interest rate soothing, in order to close the odel. Note that paraeter values are indexed in a way siilar to the notation used in input - output tables and that all equations have been expressed in linearized for around their steady state values, which are here supplied fro the outside and thus exogenously given. Taken together the considered dynaic DAS-DAD acroodel, with real wage dynaics now in the place of the noinal WPC and PPC, thus consists of the following five laws of otion: 3 uˆ = α ( ) (( ˆ uu u αur r p) ( ro )) ±αu ω lnω (3) eˆ = β ( u ) + β û (4) eu euˆ r = γ ( r r ) + γ ( pˆ ) + γ ( u ) (5) rr o rp ru ˆ ω = κ[( κ )( β ( e ) β ln ω) ( κ )( β ( u ) + β ln ω)] (6) p we wω w pu pω = β ( pˆ (7) ) representing the IS-dynaics, Okun s Law, the Taylor Rule, the dynaics of incoe distribution or of the real wage, and the updating of the inflationary cliate expression. Since steady state values are here paraeters of the odel we assue for reasons of nuerical siplicity that they are given by in the case of utilization rates and real wages and fro an annualized perspective by 0. and 0.02 as far as the steady state rate of interest and the inflation target of the central bank are concerned. lnω 3 Note that the law of otion for real wages is still given by a linear equation, if the auxiliary variable is used in the place of ω. The nonlinearities that characterize our dynaical syste are thus only due to the growth rate forulations of output and eployent dynaics.

7 We have to use in addition the following reduced for expression for the price inflation PC: 4 pˆ = κ[ β ( u ) + β ln ω+ κ ( β ( e ) β ln ω)] +, (8) pu pω p we wω which has to be inserted into the above laws of otion in various places in order to obtain an autonoous syste of differential equations in the state variables: capacity utilization u, the rate of eployent e, the noinal rate of interest r, the real wage rate ω, and the inflationary cliate expression. We have written the laws of otion in an order that first presents the dynaic equations also present in the baseline New Keynesian odel of inflation dynaics, and then the extension by our dynaics of incoe distribution and the inflationary cliate in which the econoy is operating. This odification and extension of the baseline DAS-AD odel of Asada, Chen, Chiarella and Flaschel (2006) goes beyond this earlier approach to the extent that it now also allows for positive effects of real wage changes on aggregate deand (so-called wage led regies), not yet present in the AD coponent of our original odification of the conventional AS-AD dynaics (which was always profit-led, due to the lack of an effect of incoe distribution on households consuption). Coparing later on this odel of DAS-DAD dynaics with its New Keynesian counterpart we will find fro the theoretical and epirical perspective that the laws of otion of the DAS-DAD odel iply daped oscillations if the inflation cliate is adjusting sufficiently sluggishly and if price inflation rates respond sluggishly to the excess deand on the arket for goods, while the New Keynesian is no longer deterined in the case of staggered wage and price adjustent even for active onetary policy rules (as was the case for its baseline version). The New Keynesian variant therefore needs at the least ore advanced Taylor interest rate rules than the conventional one, in order to get deterinacy and thus unique responses to disturbances of its deterinistic core dynaics. The DAS-DAD dynaics, by contrast, can becoe explosive if inflationary cliate expectations or price inflation itself are adjusting to fast. Its unboundedness ust then be taed by assuing behavioral nonlinearities at least far off the steady state that then liit the explosive nature of the dynaics such that they becoe bounded and thus econoically viable. Strategies for finding eaningful trajectories for the New and our atured Keynesian acrodynaics therefore differ radically fro each other as well as the underlying paradigs of Frisch and Keynes, characterized by persistently shocked shock absorbers on the one hand and endogenously created business fluctuations (locally explosive dynaics under certain side conditions on adjustent speeds taed by behavioral nonlinearities far off the steady state) on the other hand. 3. Siulating an estiated version of the odel The odel (3)-(7) has been estiated for the U.S. econoy ( ) on its seistructural level in Chen, Chiarella, Flaschel and Seler (2006) with the following result for estiated paraeter values, obtained fro a two-stage least-square syste estiate where the inflationary cliate was easured by a twelve quarter oving average with linearly declining weights denoted by : 2 t 4 The reduced for expression for the wage inflation PC reads: wˆ = κβ [ ( e ) β ln ω+ κ( β ( u ) + β ln ω)] +, and is of use later on when nonlinear we wω w pu pω fors of the WPC have to be considered.

8 dln u = 0. 04u 0. 94( r dln p ) 0. 54lnω t+ t t t+ t d e d u e u 08. ln t+ = 0. 8 ln t+ [or in integrated for : t = t ] r = 092. r dlnp u 00. t+ t t+ t d w e d p 2 ln t+ = 03. t 007ln. ωt ln t t dln p = 0. 04u lnω dln w t+ t t t+ t We have now one law of otion less than before, since the adaptive expectations echanis for the inflationary cliate has been replaced here by a oving average expression, which when translated back into such a echanis gives rise to an adjustent speed of approxiately β = 05. in the law of otion for the cliate. Taken together we therefore get the following nuerical specification of the reduced for 4D dynaics of Chen, Chiarella, Flaschel and Seler (2006), where the easured for of Okun s law β euˆ 02. et = ut = ut has been linearized around the steady state and inserted, and where the two 2 2 linear structural equations for dln wt+ t, dln pt+ t have been solved and deducted fro each other in order to obtain the law of otion for 2 2 dln wt+ t ( dln pt+ t ) = dln wt+ dln pt+ = dlnωt+ solely as a function of the capacity utilization rates of firs and of workers and the current level of the real wage. dln ut+ = 0. 4ut 0. 94( rt dln pt+ ) 0. 54lnωt (9) rt+ = 092. rt dlnpt ut 00. dlnωt+ = ut lnωt (0) () d t = 05(. dln pt t ) (2) + As before we have to insert the following reduced for expression for the PPC in order to get an autonoous syste of difference equations in the state variables: capacity utilization u t, the noinal rate of interest r, t the real wage rate ω t, and the inflationary cliate expression t. dln p = κβ [ ( u ) + β ln ω+ κ( β ( e ) β ln ω)] + t+ pu t pω t p we t wω t t which here siplifies to the following expression dln p = 0. t 08u + t 0. 03lnω + + t t (3) when use is ade again of Okun s law linearized around its steady state value. As noted, we have ade use of Okun s law in integrated for also in the law of otion for real wages, see eq. (6), which when inserted into it gives the following expression in front of the rate of capacity utilization ut (by which et has been replaced in the deand pressure ter of the WPC):

9 α = κ[( κ ) β β ( κ ) β ] p we euˆ w pu This paraeter α is the critical condition for the working of the so-called Rose- or real-wageeffect, since it when α is positive states that real wage growth is positively correlated with econoic activity and thus due to the estiated for of the goods arket dynaics, i.e., the law of otion for the rate of capacity utilization negatively responds to its level with a tie delay, if this law of otion for u is also taken into account. Due to these two laws of otion and size of their estiated coefficients, the role of incoe distribution in the fluctuations generated by the odel however does not appear to be an iportant one as far as the AD side of the odel is concerned, since the aggregate influence of the rate of capacity utilization on real wages is a weak one (due to the estiated for of Okun s law). Note however that though weak this feedback chain fro real wage changes via capacity utilization changes back to real wage changes is a destabilizing one, The estiated syste (9) (3) however shows that this -- fro the partial perspective -- destabilizing crossover feedback channel between real wage changes and rates of capacity utilization changes ay be stabilized by the Blanchard and Katz error correction ters in the law of otion for real wages (though this latter effect concerns the trace of the Jacobian, while the forer crossover effect concerns their inors of order two). Note however, that the dynaics of incoe distribution are here ebedded in a fraework with estiated paraeter sizes that are held constant over the whole observation period, i.e., incoe distribution here works in a Keynesian environent with in particular rigid, hence not systeatically in tie varying propensities to consue and invest, in contrast to what has been suggested by Keynes (936) in his analysis of goods arket dynaics in chapter 22. We therefore conclude that certain business cycle generators (accelerators) are still absent fro the considered dynaics in their present for which iplies that iplied phase length for the cycle will exceed considerably those actually observed for the U.S. econoy, as it is indeed exeplified in figure. We thus get fro the reduced for representation (9) (3) of our estiated DAS-DAD dynaics that the growth rates of u and ω depend both negatively in a cross-over fashion fro each other, establishing a (weakly) destabilizing feedback chain between capacity utilization and real wage dynaics, or an adverse Rose-effect, if paraeters are taken as estiated and confidence intervals still neglected. This result is due to the fact that the at first sight apparently doinant effect of e on ˆω is significantly reduced by the weak link between the rate of eployent (a stock ratio) and the rate of capacity utilization (a flow ratio) established through our estiate of Okun s law. Besides the Rose-effect we have the usual rate of interest channel in the law of otion for the rate of capacity utilization, whereby the central bank policy can influence the econoy in a stabilizing fashion, which in fact is a substitute for the conventional Keynes-effect, but whereby also the destabilizing Mundell-effect coes into being. This latter effect establishes a positive link between the rate of capacity utilization and its rate of change, since the real rate of interest depends negatively on the inflation rate and thereby also negatively on the rate of capacity utilization. However, since we have estiated that the paraeter β pu is likely to be very sall (iplying that there is no strong dependence of price inflation on deand pressure in the arket for goods), the destabilizing Mundell-effect will be relatively weak, despite a significant negative dependence of the growth rate of econoic activity on the real rate of interest. 5 5 See Chen, Chiarella, Flaschel and Seler (2005) for details on the estiated paraeters in the first two cases and the appendix in this paper for single and syste estiates in the case where a single lagged value is characterizing the backward-looking coponent in the wage-price spiral. Also the OLS estiates confir what

10 u 6 2 t = 20q Figure. Responses to positive real wage shocks for the three sets of estiated paraeter values (based on inflationary cliate ters with linearly declining weights with = 2, 6, quarter length. We have furtherore fro the partial perspective a stable dynaic ultiplier and as already noted -- a stabilizing influence of real wages on their rate of change, established by the Blanchard and Katz (999) error correction ters in the wage and price Phillips curves. There is finally a positive link between changes in the inflation cliate and the rate of capacity utilization, since the rate of price inflation and thus the inflationary cliate depend positively on the rate of capacity utilization and since again via the real rate of interest channel the rate of change of the capacity utilization of firs depends positively on the inflation cliate surrounding the current evolution of the econoy (see the reduced for (3) for the PPC as well as eq.s (2), (9)). This suppleents the findings that increasing reactions of wage inflation to deand pressure should contribute to stability, while the opposite is true for increasing reactions of price inflation to their easure of deand pressure. Of course, all these stateents are only partial in nature and ay be falsified as intuitive guidelines for the systes (in)stability if the eigenvalues of the Jacobian of the full dynaical syste are calculated nuerically, since all these feedback chains only appear in part of the inors that are to be considered in the Routh-Hurwitz conditions for local asyptotic stability. It has been established in Chen, Chiarella, Flaschel and Seler (2006), see also the next section, that the estiated sign structure always iplies local asyptotic stability of the steady state if the reaction of price inflation to deand pressure is sufficiently sall, if the inflationary cliate is updated sufficiently slow and if interest rate soothing is sufficiently weak, but the interest rate reacting to inflation gaps sufficiently strong. Taking everything together, we therefore should expect convergence back to the steady state in the case of a onetary policy that is sufficiently active -- when the considered dynaics are siulated nuerically and shocked out of their steady state position. This is indeed the case as is shown in figure (for a onetary policy that is sufficiently active) where we also show that this situation is not uch changed if our twelve quarter oving average representation of the inflation cliate is odified towards a six quarter oving average (again with linearly declining weights) and the odel reestiated. Siilar observations also hold even in the case is shown in figure, though with soewhat ore volatility than in the corresponding case of syste estiates.

11 where only one quarter is considered. i.e., when the inflation cliate is just represented by the inflation rate of the previous period as this is usually the case in theoretical analysis of the New Keynesian approach augented towards the treatent of hybrid expectations (to which the concept of an inflation regie or cliate is however then no longer associated). We thus get as a first nuerical result that the econoy sees to be very robust in the absorption of supply-, deand-side and policy shocks (if indeed the reaction of the interest rate to the inflation gap is soewhat ore active than estiated). In figure we exeplify this result through the application of a positive real-wage shock (caused by an increase in oney wages or a decline in the general price level). The response is significant decline in the rate of capacity utilization for approxiately three years and then a slow and overshooting recovery over the next seven years until the econoy starts converging back to its steady state position with ore or less ild fluctuations. This result holds unabiguously for the cliate 2 6 expressions of the discussed type, i.e., for t, t, t. However the overshooting echanis can be seen to becoe the stronger the faster the inflationary cliate adjusts to the short-run fluctuations of the actual exchange rate. It is a bit perplexing to know fro the theoretical analysis of the odel that it loses its stability by way of a Hopf bifurcation 6 if the speed of adjustent of the inflationary cliate becoes sufficiently fast and to find epirically that convergence of the dynaics back to its steady state position is guaranteed even if only a one quarter lag applies for the representation of the inflation cliate (which is then literally speaking no longer interpretable as a cliate expression). We believe that this is basically due to the fact that we did not allow in our estiation for nonlinear behavioral relationships, a task that reains to be solved. The applied econoetric ethodology, see Chen, Chiarella, Flaschel and Seler (2006) for details, and the by and large linear structure of the odel thus basically prevent in view of the bounded fluctuations contained in the eployed data set the establishent of doinant eigenvalues outside the unit circle (or with soe reservation -- with a doinant positive real part in the continuous-tie version of the odel). 7 The only weak evidence for an increased tendency towards instability is the increase in volatility that is shown in figure when the tie horizon in the foration of the inflation cliate expression becoes saller and saller. We note however again that the reaction of the Central Bank to the inflation gap ust be increased to a certain degree as copared to the estiated value in order to reove a slightly positive real eigenvalue fro the siulated continuous tie dynaics. In the eigenvalue diagras shown in figure 2 we in addition have for γ rr < γ rp -- to take note of the fact that the loss of stability by way of a Hopf-bifurcation (generally leading to the death of an unstable liit cycle around a stable corridor of the dynaics or the birth of a stable liit cycle after the loss of stability of the steady state) becoes ore and ore 2 6 delayed if we use the estiated paraeters for the cases,, in this order as shown on the left hand side of figure 2. Faster and faster adjustent of the inflationary cliate expression in our continuous tie version of the dynaics thus does not lead to a decrease in the Hopf bifurcation point, but to its increase instead. We note here that due to the estiated for of Okun s law we have that one eigenvalue of the 5D dynaics ust always be zero so that only the local instability range is clearly shown in the eigenvalue diagras in figure 2. The increased tendency towards instability if we run through the sequence 2, 6, can t t t 6 In the case where γ rr < γ rp holds, se our stability considerations below. 7 This even holds if all equations are estiated separately as single equations, see the appendix for the estiated paraeter values in this case.

12 however be irrored by our estiated odels to a certain degree when we consider the sae situation of a loss of stability by way of the paraeter β in the place of the paraeter β as is shown on the right hand side of figure 2. Here it is shown that the interval for this paraeter where stability prevails becoes saller and saller so that in the case of we even get loss of stability within the confidence interval for the estiated paraeter (see the appendix to this paper). We therefore find even in the case where behavioral nonlinearities are ignored in theory and in estiation that the vulnerability of the dynaics towards the establishent of explosive adjustent processes becoe the larger the faster the inflation cliate is adjusting in view of the actual course of price inflation. Maxiu Real Part of the Eigenvalues of the Jacobian at the Steady State pu β β pu β 6 β pu β β pu Figure 2. Eigenvalue diagras for varying paraeter sizes. We have found out in addition in Chen, Chiarella, Flaschel and Seler (2006) that all partial feedback chains (including the working of the Blanchard and Katz error correction ters) translate theselves into corresponding noral eigenvalue reaction patterns for the full 5D dynaics (with Okun s law added in its estiated derivative for), with the exception of the speed paraeter β w, where the eigenvalue analysis has shown that increasing wage flexibility ay indeed becoe destabilizing if it is ade very large. This provides one exaple for the situation that partial econoic insight can be isleading due to the fact that the corresponding feedback chain is only a sall coponent of the any inors of the Jacobian of the dynaics at the steady state that have to be investigated in the application of the Routh-Hurwitz conditions to the full 4D dynaics (where Okun s law is applied in level

13 for). Increasing price flexibility has been found to be destabilizing, since the growth rate ê of econoic activity can thereby be ade to depend positively on its level (via the real rate of interest channel, see eq. (3)), leading to an unstable augented dynaic ultiplier process in the trace of the Jacobian J of the syste under such circustances. Furtherore, such increasing price flexibility will give rise to a negative dependence of the growth rate of the real wage on econoic activity (whose rate of change in turn depends negatively on the real wage) and thus lead to further sign changes in the Jacobian J. Increasing price flexibility is therefore bad for the stability of the considered dynaics fro at least two perspectives. Nevertheless as the odel is estiated there sees to be no proble for the working of the econoy, since econoic shocks ay have long lasting consequences (due to our easureent of tie-constant paraeters) when sufficiently large, but are always absorbed by the econoy through nearly onotonic adjustents, once the effect of the shock has becoe reversed. r the estiated odel 'pi2' : =.02 econoic breakdown due to a too low inflation target : =.003 γ =.5 rp γ = ru tie Figure 3. Downward oney wage rigidity and too restrictive inflation targets. This ipression ay however be isleading if one further aspect of the functioning of actual arket econoies is taken into account (that has not yet been paid attention to in our estiation procedures). Hoogenveen and Kuipers (2000) have established for six European countries that oney wages are not only copletely rigid downwards, but have in fact already a floor for their rate of growth, which is bounded fro below by a positive value. They basically therefore establish a kink in the WPC already at positive rates of wage inflation. Chen and Flaschel (2005) do not find such a strong result in the case of the U.S. econoy, but find also at least soe evidence that oney wages can be considered as being downwardly rigid. We thus now reconsider the above dynaical syste for the new situation where wages can rise as specified by it, but cannot fall, i.e., we exclude wage deflation now fro consideration. In such a case we can establish the results shown in figure 3 where the estiated odel and its shock absorber properties are repeated for the case 2 (with an

14 inflation target of the central bank of again two percent). Adding coplete downward oney wage rigidity to the convergent dynaics exeplified in figure 2 (and a onetary policy that is tighter in its inflation target: = ) now however iplies a radical change in the syste s behavior. Since oney wages cannot fall and since price can still fall in the depression generated by the assued positive real wage shock though price flexibility was easured only very sluggishly we get in such a situation that the real wage ust increase. This downward adjustent echanis continues to work and it akes the depression deeper and deeper until the econoy breaks down, exeplified in figure 3 by eans of the evolution of the noinal rate of interest. There exists therefore a great danger for systes with profit-led goods arket dynaics, since downward priceflexibility coupled with downwardly rigid wages will then necessarily lead the econoy into a deflationary spiral when deflation begins to start through shocks or other events. We also show in this figure how a ore active interest rate policy with respect to the inflation as well as the output gap can avoid such a breakdown if it is chosen sufficiently strong. For our purposes we however here only need that a global floor to the evolution of the oney wage (or its inflation rate) can be disastrous in a situation that initially appeared to be a very stable one, without such a nonlinearity. The question therefore is by which echaniss such a onotonic tendency towards ore and ore severe depressions can be stopped or even reversed. this question will be further pursued after the following section which is devoted to a stability analysis of the dynaics with the estiated paraeter signs and the theoretical occurrence of instability if a floor to oney wage inflation is added to the odel. 4. Analyzing an estiated version of the odel Based on our estiates (9) (2), the theoretical odel (3) (8) can now be siplified to the following qualitative forat (with b = β eu ˆ for notational siplicity): uˆ = α ( ) ( ˆ uu u αur r p ( ro )) αu ω lnω (4) r = γ ( r r ) + γ ( pˆ ) + γ ( u ) (5) rr o rp ru b ˆ ω = κ[( κ )( β ( u ) β ln ω) ( κ )( β ( u ) + β ln ω)] (6) p we wω w pu pω = β ( pˆ (7) ) b pˆ = κ[ β ( u ) + β ln ω+ κ ( β ( u ) β ln ω)] +, (8) pu pω p we wω where the ˆp equation has to be inserted again in soe of the other equations in order to arrive at an autonoous syste of differential equations. Doing this and rearranging ites then gives rise to, when linearized around the steady state: uˆ = [ α κ( β + κ β b)] u α r [ α α κ( β κ β )] ω+ α + const uu pu p we ur uω ur pω p wω ur = ω+ ± au ar 2 a3 a4 ao r = [ γ κ( β + κ β b) + γ ] u γ r+ γ κ( β κ β ) ω+ γ + const rp pu p we ru rr rp pω p wω rp =+ ± ω+ ± bu br 2 b3 b4 bo ˆ ω= κ([( κ ) β b ( κ ) β ] u [( κ ) β + ( κ ) β ] ω)] + const =+ cu cω ± c 3 3 p we w pu o p wω w pω = β κ[( βpu+ κpβweb) u+ κ( βp ω κpβw ω) ω] + const =+ du± dω ± d o

15 Making use of our estiated paraeter values we have assued, on the one hand, in the law of otion for the rate of capacity utilization u that the pˆ ˆ u, p ω coponents are doinated by the direct influences of u, ω on the growth rate of capacity utilization. In the law of otion for real wages we, on the other hand, we assue however, by way of a slightly larger coefficient b, that the growth rate of real wages depends positively on the rate of capacity utilization (i.e., that β we is the doinant ter in this respect). This positive dependence ay be a weak one, and also as before weakly negative, since our estiate of Okun s law iplies in any case only a fairly weak ipact effect of the capacity utilization rate on the rate of eployent. We thereby get a sign structure for the partial derivatives of our dynaical syste and thus its Jacobian J at the steady state, as it is shown below (where the ± ites are solely due the two opposing real wage or Blanchard and Katz error correction ters in the reduced for PPC). J + + ± + = ± 0 Proposition : Assue that the sign structure of the atrix J applies and that its entry J 23 is sufficiently sall. 8 Then: The steady state of the dynaics (4) (7), with (8) inserted into the, is locally asyptotically stable, if the inflationary cliate expression is updated in a sufficiently sluggish way, and if γ rr < γ rp holds true. Proof: Let us first consider the case where β = 0 holds true. We consider the subatrix J(3,3) for the reaining laws of otion, with J (3, 3) = The characteristic polynoial of this atrix, is given by: J 23 set equal to zero, which is then given by: p( λ) = λ + kλ + k λ + k, k = trace J(3, 3), k = det J(3, 3) and it is easily shown to have only positive coefficients. Furtherore, the condition kk k > is also fulfilled, since det J (3, 3) is copletely doinated by the expressions that ake up kk 2. We thus have that the Routh-Hurwitz conditions for local asyptotic stability apply to the given situation, iplying that the real parts of the eigenvalues of this polynoial ust all be negative. Consider now the case β > 0. According to our assuptions we then get for de t J the sign 8 Corresponding to our estiate of the paraeter sizes of the Blanchard and Katz error correction ters in the reduced-for PPC.

16 structure: det J = det = det which iplies a positive deterinant due to our assuption on the relative sizes of the paraeters γ rr, γ rp. Making the paraeter β slightly positive oves the eigenvalues with three negative real parts and one zero eigenvalue slightly, such that they are now all in the negative half plane of the coplex plane (since eigenvalues depend continuously on the paraeters of the odel and since the deterinant is the product of all four eigenvalues). This iplies the assertion of proposition. Proposition 2: Assue that the sign structure of the atrix is sufficiently sall. applies and that its entry J 23 J (i) Increasing the adjustent speed β pu, β wu to a sufficient degree gives rise to a Hopf-bifurcation via dynaic ultiplier instability where the syste up to flukes looses its asyptotic stability after the death of an unstable liit cycle or the birth of a stable liit cycle. (ii) Assue α = 0 in which case (i) does not apply. Then: Increasing the adjustent u r speed β to a sufficient degree gives again rise to a Hopf-bifurcation towards the pu above type of instability, now via an adverse real wage adjustent (an adverse Rose effect). (iii) Increasing the adjustent speed β to a sufficient degree gives also rise to the above type of a Hopf-bifurcation, now via an adverse real interest rate adjustent (the so-called Mundell-effect). Proof: (i) In this case the entry J, characterizing the overall effect of utilization changes on the growth rate of capacity utilization, becoes positive and can be ade as large as needed in order to arrive at a positive trace of the atrix J. The Hopfbifurcation then occurs when kk 2 k3 becoes zero, which ust be the case before trace J = 0 is established. (ii) In this case reains negative, while J J is zero. The only entry in the coefficient where J 3 J 2 2 k2 that then depends on the paraeter β pu is then given by J 3 J 3, ust becoe positive for increasing β, while reains negative, thereby establishing a positive feedback channel between capacity utilization and real wages that akes k2 negative if β pu becoes sufficiently large. The Hopfbifurcation then occurs when kk 2 k 3 becoes zero, which ust be the case before k 2 = 0 is established. pu J 3

17 (iii) Obvious, since the paraeter β only appears in the product J which 4J4 establishes a positive link between the evolution of the inflation cliate and capacity utilization u, the destabilizing Mundell-effect of conventional acrodynaic odel building. Proposition 3: Assue that the sign structure of the atrix J applies. Assue furtherore that onetary policy is ipotent by setting α ur = 0. Assue finally that the negative Blanchard and Katz error correction echanis in the real wage dynaics is sufficiently weak (i.e., the ter ( κ ) β is chosen sufficiently sall). Then: w pω The steady state of the considered dynaical syste unstable in the downward direction, if there is a global floor to oney wage inflation at its steady state value, i.e., any initial and contractive u or ω shock will then lead to an ever accelerating contraction of the econoy. Proof: In the considered situation, the interacting dynaics are reduced to a 2D dynaical syste in the rates u, ω. This syste exhibits a negative deterinant of its Jacobian at the steady state uo =, ωo = and two D stable anifolds that cannot be reached by contractions in u or expansions in ω. This iplies that such shocks always lead to trajectories with a declining rate of capacity utilization along the. If onetary policy is only weakly influencing the private sector and if the Blanchard and Katz real wage error correction echanis in the PPC is weak, we thus have that the reaction of price levels to their corresponding deand pressure ite and the issing reaction of wages in this regard will allow for an adverse adjustent. i.e., an increase in real wages that will lead the econoy into deeper and deeper depressions. The question then is whether interest rate effects in goods deand and interest rate steering by the Central Bank can help to avoid such an outcoe, since of course the destabilizing Mundell effect will then also be back in the feedback interactions of our econoy, or whether onetary policy needs a systeatic overhaul in a situation where there is an adverse real wage effect at work. Our analytical findings in the estiated situation thus are that the econoy ay work like a shock absorber for certain ranges of its paraeter values, but that this property ay get lost in a variety of ways which then deand for the introduction of further behavioral nonlinearities that can keep the resulting dynaics a bounded one despite the existence of centrifugal forces around its steady state position. Let us briefly copare this situation with the New Keynesian acrodynaic odel and its solution ethodology. We have already shown in section 2 that the New Keynesian wageprice dynaics can be reduced to the following two laws of otion for the tie rate of change of the wage and the price inflation rate: w p = = βwy y + βw ωθ β y β θ where θ denotes the log of the real wage ω. With respect to the standard output dynaics of the New Keynesian approach ( y = lny, Y = ): py pω y y = α ( r r ) p t+ t yr t t+ 0

18 one furtherore gets when also transferred into continuous tie: = p y α [( β ) + ( β + β ) y+ β ωθ ) (9) yr r ry py p by assuing in addition and by inserting into the the following conventional type of a onetary policy rule r = r + β + β y. (20) t o r ry In view of these three laws of otion there reains finally the law of otion for real wages to be deterined, which due to θ = lnω here siply reads (2) w p θ = We thus get in a coparable New Keynesian baseline situation with only forward-looking behavior in wage, price and output foration an autonoous linear dynaics, now in the w variables, p, y and θ for which following the New Keynesian rational expectations solution ethod the zone of deterinacy is to be deterined. In the present situation with four forward-looking variables this eans in striking contrast to our proposition that we have to find situations (if necessary, by odifying the eployed type of Taylor rule) where all roots have positive real parts, since we then and only then have a single bounded solution and thus deterinacy in the reaction of the econoy for the considered dynaical syste. In the deterinistic core of the New Keynesian odel with both staggered price and wage setting we thus get that the only possible response of the syste to isolated shocks is to return instantaneously to its steady state position which is here given by ( 0000),,,. The situation therefore is here such that we need an anti-routh-hurwitz theore, i.e., a theore that iplies that all roots of the Jacobian of the dynaics have positive real parts. 9 Then (and only then) is there a unique response of the dynaics that satisfies the atheatical side-condition that responses of the dynaics to isolated shocks ust be bounded ones. Systes having ultiple bounded solutions are norally blocked out fro further consideration and thus not considered as being useful for the analysis of Keynesian business cycle theory. In the present situation this eans that though price and wages are subject to staggered contracts there is no inertia in the adjustent of wages and price to isolated shocks, in stark contrast to our alternative proposal of a odel of Keynesian acrodynaics. The Jacobian of the 4D dynaical syste under consideration reads J 0 0 βwy βw ω 0 0 βpy β pω =. 0 αyr( βr ) αyr( βry + βpy ) αyrβp ω The forulation of such a theore is not difficult, since a characteristic polynoial p( λ) = λ + kλ + k λ + k λ + k, with only positive real parts of eigenvalues is exactly given when p( λ) = λ kλ + k λ k λ +k fulfills the conditions of the Routh-Hurwitz theore in its standard 4 for. This in particular iplies that the deterinant of the considered Jacobian ust be positive as well as the su of principal inors of order three, i.e., the su of the deterinants of the 3D sub-atrices of this Jacobian along the diagonal of it.

19 For the deterinant of this Jacobian we therefro get > > J = ( βwyβp ω + βpyβw ω) αyr ( βr ) 0 iff βr < < We thus iediately get that an active onetary policy rule ( β r > ) does not allow for deterinacy, since roots with only positive real parts iply that the deterinant of the Jacobian (the product of these roots) ust be positive. If, on the other hand, we have βr, we only have to consider the su of principal inors of order 3 for a Jacobian with the sign structure J = This su is then given by: which is easily shown to be negative, since it can be shown that the first deterinant can be reduced to We thus get that the Routh-Hurwitz coefficient in front of λ is positive which violates the Routh-Hurwitz conditions for total instability, see the preceding footnote for details. We thus get as a rearkable analytic result that the advanced baseline odel of New Keynesian acrodynaics which is coparable to our Keynesian odel with its advanced wage-price spiral is copletely indeterinate and thus to be blocked out fro consideration. As was shown, this holds even if the conventional Taylor rule is eployed which thus is incapable of enforcing deterinacy, in striking contrast to the New Keynesian baseline odel with only staggered price setting. There is thus no workable counterpart to the atured, but conventional Keynesian dynaics we have investigated in this section, which was shown to allow for the Frisch paradig for estiated paraeter values, but which can also easily give rise to probleatic scenarios if certain speeds of adjustent becoe larger or if there is a global floor to the downward adjustent of noinal wages that is sufficiently tight. We clai, but do not prove this here that this result also holds in discrete tie, under reasonable assuptions on the sizes of the eployed paraeter values.

20 In such situations our approach deands further behavioral nonlinearities in the case of explosive downward (or upward) business fluctuations, but not for the iposition of a atheatical boundedness condition that if deterinate keeps the dynaics always in its steady state position, a result that is not at all in line with Keynes (936) own analysis of the trade cycle echanis, see his chapter 22 and the uses he there akes of his three central paraeters, the arginal propensity to consue, the arginal efficiency of investent and the state of liquidity preference. However, concerning our own approach, we find that the results on various degrees of downward oney wage rigidities do not unabiguously support Keynes view that workers resistance against oney wage reductions is always good for econoic stability. Concentrating on onetary policy solely, the question in our case is whether a policy rule can be found that can tae the outbreak of accelerating downward or upward wage-price spirals and other explosive feedback echaniss, while in the New Keynesian case one has to odify the dynaics such that there is a single bounded response in the considered 4D case or in higher diensions if interest rate policy gives rise to such an increase in diension. 5. Midrange downward oney wage rigidity In this section we show by eans of a nuerical exaple that the probleatic downward rigidity of oney wages ay to soe extent be needed in order to stabilize the econoy when it is subject to explosive fluctuations caused by an increase in the speed with which the inflationary cliate expression is adjusted. This rigidity should not however be global in nature, but give way again to downward wage flexibility if the rate of eployent becoes sufficiently low. This particular, not iplausible ix of different wage inflation regies is a bit surprising with respect to its stability iplications, but coes at least not copletely unotivated, due to the fact that wage flexibility tends to be stabilizing and price flexibility destabilizing fro the partial perspective of the real wage channel discussed above. Figure 4. Three possible regies for wage inflation.

21 In order to foralize the envisaged three regies of wage inflation needed for our subsequent siulations we ake use of the three alternatives shown in figure 4 for our reforulation of the WPC, fro which the dynaics of real wages and price inflation in their reduced for presentation ust now be derived. Underlying these three situations is the assuption that wages behave as in the original odel considered in sections 2 and 3 when their rate of change is above a certain floor f and when the rate of eployent is above a certain critical level e below which workers will again accept faster decreases in their oney wages than just the level f. Wage inflation is in the latter case assued to be driven by wˆ = β ( e ) + κ pˆ + ( κ ) we w w in the place of the working of the Phillips curve in noral or overheated situations: wˆ = β ( e ) β ln ω+ κ pˆ + ( κ ). we wω w w In between we will find as stated a regie where wage inflation (or deflation) is just given by a rate f. These three scenarios translate theselves into reduced-for real wage and price level dynaics as follows, where we now denote by wˆ red the reduced for oney wage Phillips curve of the original odel: If ˆ red f e e w <,, then ˆ = κβ [ e β ln ω+ κ( β u+ β ln ω)] + wred we wω w pu pω ˆ ω = ( κp)( f ) βpuu+ β p ω lnω (22) pˆ = + κ ( f ) + β u+ β ω lnω, (23) p pu p if ˆ red f e e w,, then the original dynaics (6), (8), applies while in all other situations we have to apply the equations: ˆ ω = κ[( κ p) βwee ( κw)( βpuu+ βp ω ln ω)] (24) pˆ = κβ [ u+ β ln ω+ κβ ( e] +. (25) pu pω p we The oney wage behavior underlying these odified dynaics is suarized in figure 4. This situation represents an appropriate odification of Filardo s (998) epirical analysis of such a type of Phillips curve, where however the price inflation gap (with respect to expected inflation) is used on the vertical axis. There is thus soe evidence for such a three regies PC, though in a soewhat different context still. When one applies this type of a nonlinear WPC to our sei-structural odel in annualized for, here fro this perspective with a tight inflation target of the central bank of = , one gets the hierarchy of events shown in figure 5: In situation, where there is no nonlinearity in the WPC, we still have the convergent result of figure 2 (a positive real wage shock of 4 percent is here applied), though indeed the inflation target is now soewhat below the floor to wage inflation. In situation 2, we have that the floor f = 00. applies

22 globally and get again econoic breakdown due to the adverse working of the Rose real wage effect. Using in situation 3 then that wage flexibility becoes reestablished again with the sae paraeter value β we, already below eployent rates e = 099., we get an interediate situation in which the rate of eployent stays below 0.99, the rate of capacity utilization converges approxiately to the value 0.95, and where there is ongoing deflation with the rate The real wage however stays 2 percent above its original steady state value and the noinal rate of interest reains positive, but is close to zero. The long-run of the odel therefore departs significantly fro the steady state values of the unrestricted dynaics with its linear WPC. We therefore fro this exaple that the three regie WPC is better as the one with a global floor to oney wage inflation, but the copletely unrestricted odel with its linear WPC provides still the best outcoe after a contractionary real wage shock has hit the econoy in these three scenarios. f =-, e= - f =-, e =- f =.0, e =.99 f =.0, e =.99 f =.0, e =- f =.0, e = - tie tie Figure 5. Convergent unrestricted dynaics and the role of piecewise nonlinear WPC s ( = ) The exaple just discussed applies to the situation where the adjustent of the inflationary cliate is still sufficiently sluggish to guarantee the stability properties shown in figure. The obtained results thus characterize an econoy that exhibits strong convergence back to the originally given steady state if not restricted by behavioral nonlinearities of the discussed type. Let us next investigate a situation where the econoy is destabilized by a change in the speed of adjustent of the inflationary cliate that is surrounding it. We now assue in the place of the value 0. 5 the value 5. 2 for the paraeter β in the considered situation. The result, in ters of capacity utilization, is shown in figure 6 by the cyclical tie series with syetrically and rapidly increasing aplitudes of the cycle. The figure 6 therefore shows in the copletely unrestricted case a tie series for the rate of capacity utilization that will sooner or later lead to econoic collapse if there is no change in the behavior of the econoy. Adding now a global floor of f = to the dynaics in their WPC coponent does not iprove this situation, but leads again to onotonic econoic breakdown instead. However if we allow for a third regie as described in figure 4 on its left hand side, we then get evolutions of the state variables of the odel that reain bounded to

23 an econoically eaningful doain. In addition, these dynaics are now of a atheatically coplex type (but only soewhat irregular fro the econoic point of view) as is shown in Chen, Chiarella, Flaschel and Seler (2006) by investigating the stable trajectory shown in figure 6 in ore detail. Figure 6. The role of regie changes in wage inflation dynaics With the assued change in the adjustent speed of the inflationary cliate expression the econoy is therefore no longer viable in the long run (but cyclically explosive) and it becoes even less viable if a global floor f = is introduced into the estiated WPC as shown in figure 5. Yet assuing a WPC as discussed in connection with figure 4 overcoes not only this latter onotonic downturn, but also the explosive fluctuations of the unrestricted case. Soe downward flexibility of oney wages in a certain idrange interval, giving way however to downward flexibility of oney wages again already at two percent rate of uneployent here provides viability to the evolution of the trajectories of the dynaics as indicated in figure 6, here over a 50 year horizon. 6. Wage-led econoies, increasing adjustent speeds and the eergence of coplex dynaics In this section we provide soe further siulation of the general odel with adissible, but no longer estiated paraeter values in order to also consider in particular a situation where the Rose effect works in the opposite way, i.e., where aggregate deand and the adjustent of the rate of capacity utilization depend positively on the real wage and where therefore wage flexibility with respect to deand pressure on the labor arket should be destabilizing. In such a situation a global floor to wage inflation or deflation should therefore save the econoy fro econoic breakdown, since downwardly rigid oney wages cobined with downwardly (soewhat) flexible prices leads to real wages increases which in the present situation stiulates econoic activity and thus should lead the econoy out of the

24 depression. Yet, due to the fact that the unrestricted econoy is here in a strong way an explosive one, we get after each recovery that explosive forces coe into being each tie in a soewhat odified anner, until the econoy falls back again into a depression with downward oney wage rigidity avoiding its further destabilization until a new recovery sets in. The base paraeter set underlying the siulations shown below is the following one: β =, β = 04., κ = 03., β = 0. 8, β = 04., κ = 0. 7, β = 0. 5, pu pω p we wω w αuu = 022., αu ω = 0., αur = 025., βeu = 05., βeuˆ = 05., βe ω = 05., γ = 0., γ = 0. 5, γ =, f = , e= 0, ω =. 0. rr rp ru shock Speeds of price and wage adjustent are soewhat higher now, while the adjustent speed of the inflationary cliate is only /3 of the value used in the preceding section. We have a positive real wage effect in the goods arket dynaics (a wage-led regie now) and have now also included a negative real wage effect in the law of otion of the rate of eployent, which furtherore now depends on the level of the rate of capacity utilization in addition to the growth rate of capacity utilization we have used so far as the sole deterinant of rate of eployent changes. The rate of eployent is thus no longer strictly positively correlated with the rate of capacity utilization as was the case in our estiate of the odel (which akes the critical α ) condition considered in section 3 ore difficult to obtain, though of course wage flexibility ust now be destabilizing). Finally, onetary policy is now ore active with respect to the state of the business cycle and we have a floor to wage deflation that is practically zero. The result of this cobination of paraeter values will be, as is shown below, that the nonlinear WPC of figure 2, now with a global floor ( e = 0 ) in fact the only iportant nonlinearity in our 5D dynaical syste is capable of keeping a highly explosive unrestricted dynaics within econoically eaningful bounds. This result is achieved in a way that akes the resulting attractors coplex fro the atheatical perspective, though not too irregular fro the econoic perspective. We illustrate the coplex dynaics that is generated by this specific paraeter set at first by showing in figure 7 projections of its now 5D forat (since the rate of eployent is now oving independently fro the rate of capacity utilization to soe extent) into 2D subplanes of the full 5D phase space. We see top left in figure 7 the partial phase plot of the real wage against the rate of eployent with a by and large clockwise orientation and phase length corresponding to what is known on the Goodwin (967) 2D growth cycle odel and its epirical analog. In periods of high eployent however this clockwise oveent of these two state variables gives way to soe local and fast fluctuations which represent the explosive part of the dynaics. Below this figure we see the projection of the attractor into the rate of eployent / inflation cliate subspace where we would expect due to what is known fro uneployent - inflation phase plots (and their clockwise orientation) a counterclockwise orientation which is not clearly visible there. This orientation is however typical for the projection of the attractor top right, there for a capacity utilization rate and inflation cliate subspace projection. We see there too, that when recovery sets in, the dynaics are squeezed through a sall corridor or eye of a needle followed by a sall stagflation cycle, which is then followed by a large cycle until the econoy is squeezed back into the sall corridor for its next upswing phase. The tie series botto-right adds to this the inforation that the sequence of business fluctuations generated by the present paraeter set is irregular and

25 regular at one and the sae tie, irregular fro the atheatical point of view in its aplitudes and regular fro the econoic point of view due to the repetitive behavior in the succession of sall and large cycle patterns. e ω u u e t = 000 t= 2000 Figure 7: Projections and a tie series representation of the attractor of the dynaics u u α uw γ ru ω.0 r β we β pu Figure 8: Bifurcation diagras for selected adjustent speeds and behavioral coefficients of the dynaics